Introduction and Model Setup
The HJB System and Verification
Numerical Results
Worst-Case Portfolio Optimization
in a Market with Bubbles
Christoph Belak
Department of Mathematics
Kaiserslautern University of Technology
Germany
School of Mathematical Sciences
Dublin City University
Ireland
Joint work with Sören Christensen (CAU Kiel) and Olaf Menkens (Dublin City University)
6th AMaMeF Conference, Warsaw, Poland
June 11, 2013
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Overview
1
Introduction and Model Setup
2
The HJB System and Verification
3
Numerical Results
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Introduction
Worst-Case Portfolio Optimization in a Market with Bubbles
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Introduction
Worst-Case Portfolio Optimization in a Market with Bubbles
We study
1
an optimal investment problem, in which the investor aims to maximize
expected utility from terminal wealth,
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Introduction
Worst-Case Portfolio Optimization in a Market with Bubbles
We study
1
an optimal investment problem, in which the investor aims to maximize
expected utility from terminal wealth,
2
while assuming that bubbles may be present in the market which may lead
to crashes,
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Introduction
Worst-Case Portfolio Optimization in a Market with Bubbles
We study
1
an optimal investment problem, in which the investor aims to maximize
expected utility from terminal wealth,
2
while assuming that bubbles may be present in the market which may lead
to crashes,
3
and we assume that the investor takes a worst-case perspective towards
the impact of these crashes.
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Introduction
Worst-Case Portfolio Optimization in a Market with Bubbles
We study
1
an optimal investment problem, in which the investor aims to maximize
expected utility from terminal wealth,
2
while assuming that bubbles may be present in the market which may lead
to crashes,
3
and we assume that the investor takes a worst-case perspective towards
the impact of these crashes.
The Worst-Case Problem
h
i
sup inf E U(XTπ,θ ) .
π
θ
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Literature
Literature related to Bubbles:
Loewenstein and Willard (2000), Cox and Hobson (2005), Jarrow, Protter and
Shimbo (2007, 2010), Biagini, Föllmer and Nedelcu (2013), ...
⇒ Compare with Föllmer’s talk.
Literature related to the Worst-Case Approach:
Korn and Willmot (2002), Korn and Menkens (2005), Korn and Steffensen
(2007), Seifried (2010), ...
⇒ Compare with Menkens’ talk.
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Market
We start with a Black-Scholes market:
dBt = 0,
dSt = αSt dt + σSt dWt .
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Market
We start with a Black-Scholes market:
dBt = 0,
dSt = αSt dt + σSt dWt .
Let Zt be an observable continuous-time Markov chain with finite state space
{0, 1, . . . , d}.
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Market
We start with a Black-Scholes market:
dBt = 0,
dSt = αSt dt + σSt dWt .
Let Zt be an observable continuous-time Markov chain with finite state space
{0, 1, . . . , d}.
State 0 corresponds to a market regime without a bubble – no crash may
occur.
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Market
We start with a Black-Scholes market:
dBt = 0,
dSt = αSt dt + σSt dWt .
Let Zt be an observable continuous-time Markov chain with finite state space
{0, 1, . . . , d}.
State 0 corresponds to a market regime without a bubble – no crash may
occur.
State i ∈ {1, . . . , d} corresponds to a regime in which a bubble is
present. The bubble may burst, leading to a crash of maximum relative
size κi , i.e.
Sτ = (1 − κ)Sτ − ,
0 ≤ κ ≤ κi .
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Market
We start with a Black-Scholes market:
dBt = 0,
dSt = αSt dt + σSt dWt .
Let Zt be an observable continuous-time Markov chain with finite state space
{0, 1, . . . , d}.
State 0 corresponds to a market regime without a bubble – no crash may
occur.
State i ∈ {1, . . . , d} corresponds to a regime in which a bubble is
present. The bubble may burst, leading to a crash of maximum relative
size κi , i.e.
Sτ = (1 − κ)Sτ − ,
0 ≤ κ ≤ κi .
After a crash, Zt is assumed to jump back to state 0.
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Bubbles and Crashes
Zt in state 0
,→ No crash possible
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Bubbles and Crashes
Zt in state 0
,→ No crash possible
↓
Zt jumps to state i
,→ Investor receives warning
,→ Crash of maximum size κi possible
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Bubbles and Crashes
Zt in state 0
,→ No crash possible
↓
Zt jumps to state i
,→ Investor receives warning
,→ Crash of maximum size κi possible
.
Crash (τ, κ) occurs
,→ Stock price crashes
by a fraction of κ
,→ Zt jumps back to
state 0
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Bubbles and Crashes
Zt in state 0
,→ No crash possible
↓
Zt jumps to state i
,→ Investor receives warning
,→ Crash of maximum size κi possible
.
&
Crash (τ, κ) occurs
Zt jumps to state j
,→ Stock price crashes
by a fraction of κ
,→ Zt jumps back to
state 0
,→ Investor receives
new information
,→ Crash of maximum
size κj possible
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Investor
In each state i = 0, . . . , d, the investor can choose which fraction πti of her
total wealth Xt to invest in the risky asset.
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Investor
In each state i = 0, . . . , d, the investor can choose which fraction πti of her
total wealth Xt to invest in the risky asset.
Given a crash scenario θ = (τk , κk )k∈N , the total wealth then evolves as
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Investor
In each state i = 0, . . . , d, the investor can choose which fraction πti of her
total wealth Xt to invest in the risky asset.
Given a crash scenario θ = (τk , κk )k∈N , the total wealth then evolves as
X0 = x,
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Investor
In each state i = 0, . . . , d, the investor can choose which fraction πti of her
total wealth Xt to invest in the risky asset.
Given a crash scenario θ = (τk , κk )k∈N , the total wealth then evolves as
X0 = x,
dXt = απti Xt dt + σπti Xt dWt ,
Christoph Belak, Sören Christensen, Olaf Menkens
on {Zt = i},
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Investor
In each state i = 0, . . . , d, the investor can choose which fraction πti of her
total wealth Xt to invest in the risky asset.
Given a crash scenario θ = (τk , κk )k∈N , the total wealth then evolves as
X0 = x,
dXt = απti Xt dt + σπti Xt dWt ,
Xτk = (1 −
πτi k κk )Xτk − ,
Christoph Belak, Sören Christensen, Olaf Menkens
on {Zt = i},
on {Zτk − = i}.
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Investor
In each state i = 0, . . . , d, the investor can choose which fraction πti of her
total wealth Xt to invest in the risky asset.
Given a crash scenario θ = (τk , κk )k∈N , the total wealth then evolves as
X0 = x,
dXt = απti Xt dt + σπti Xt dWt ,
Xτk = (1 −
on {Zt = i},
πτi k κk )Xτk − ,
on {Zτk − = i}.
We say that a strategy π = (π 0 , . . . , π d ) is admissible, if it leads to nonnegative
wealth for all possible crash scenarios θ = (τk , κk )k∈N . This holds if
πi ≤
1
,
κi
Christoph Belak, Sören Christensen, Olaf Menkens
for all i.
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Worst-Case Problem
The aim of the investor is to find the strategy π ∗ = (π 0,∗ , . . . , π d,∗ ) which
performs best if the worst-possible crash scenario θ = (τk , κk )k∈N occurs.
The Worst-Case Problem
h
i
V (t, x, i) = sup inf E(t,x,i) U(XTπ,θ ) .
π
Christoph Belak, Sören Christensen, Olaf Menkens
θ
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The Worst-Case Problem
The aim of the investor is to find the strategy π ∗ = (π 0,∗ , . . . , π d,∗ ) which
performs best if the worst-possible crash scenario θ = (τk , κk )k∈N occurs.
The Worst-Case Problem
h
i
V (t, x, i) = sup inf E(t,x,i) U(XTπ,θ ) .
π
θ
Here, U is assumed to be the power utility function:
U(x) =
1 γ
x ,
γ
Christoph Belak, Sören Christensen, Olaf Menkens
γ < 1, γ 6= 0.
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Overview
1
Introduction and Model Setup
2
The HJB System and Verification
3
Numerical Results
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The HJB Equation for i = 0
HJB Equation for i = 0
The value function V (·, ·, 0) and the corresponding optimal strategy in state 0
can be determined by solving the following HJB equation:
d
X
0 = sup Lπ V (t, x, 0) +
q0,j V (t, x, j) .
π
Christoph Belak, Sören Christensen, Olaf Menkens
j=0
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The HJB Equation for i = 0
Denote by (qi,j )0≤i,j≤d the generator matrix of Zt and let
Lπ V = Vt + απxVx +
1 2 2 2
σ π x Vxx .
2
HJB Equation for i = 0
The value function V (·, ·, 0) and the corresponding optimal strategy in state 0
can be determined by solving the following HJB equation:
d
X
0 = sup Lπ V (t, x, 0) +
q0,j V (t, x, j) .
π
Christoph Belak, Sören Christensen, Olaf Menkens
j=0
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Verification for i = 0
Using the HJB equation
d
X
0 = sup Lπ V (t, x, 0) +
q0,j V (t, x, j) ,
π
j=0
it is easy the verify that the optimal strategy for state 0 is simply the Merton
fraction
α
πt0,∗ =
.
(1 − γ)σ 2
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Verification for i = 0
Using the HJB equation
d
X
0 = sup Lπ V (t, x, 0) +
q0,j V (t, x, j) ,
π
j=0
it is easy the verify that the optimal strategy for state 0 is simply the Merton
fraction
α
πt0,∗ =
.
(1 − γ)σ 2
The value function is given by
V (t, x, 0) =
1 γ 0
x f (t),
γ
where f 0 (t) solves
d
ft0 (t) = −
X
γα2
f 0 (t) −
q0,j f j (t),
2
2(1 − γ)σ
j=0
Christoph Belak, Sören Christensen, Olaf Menkens
f 0 (T ) = 1.
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The HJB Equation for i > 0
Define the following sets:
n
o
A1 := π : V (t, x, i) ≤ V (t, (1 − κi πt )x, 0) ,
The value function V (·, ·, i) and the corresponding optimal strategy in state i
can be determined by solving the HJB system
d
i
h
X
qi,j V (t, x, j) ,
0 ≤ sup Lπ V (t, x, i) +
π∈A1
j=1
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The HJB Equation for i > 0
Define the following sets:
n
o
A1 := π : V (t, x, i) ≤ V (t, (1 − κi πt )x, 0) ,
d
o
n
X
A2 := π : Lπ V (t, x, i) +
qi,j V (t, x, j) ≥ 0 .
j=1
The value function V (·, ·, i) and the corresponding optimal strategy in state i
can be determined by solving the HJB system
d
i
h
X
qi,j V (t, x, j) ,
0 ≤ sup Lπ V (t, x, i) +
π∈A1
j=1
h
i
0 ≤ sup V (t, (1 − κi πt )x, 0) − V (t, x, i) ,
π∈A2
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
The HJB Equation for i > 0
Define the following sets:
n
o
A1 := π : V (t, x, i) ≤ V (t, (1 − κi πt )x, 0) ,
d
o
n
X
A2 := π : Lπ V (t, x, i) +
qi,j V (t, x, j) ≥ 0 .
j=1
The value function V (·, ·, i) and the corresponding optimal strategy in state i
can be determined by solving the HJB system
d
i
h
X
qi,j V (t, x, j) ,
0 ≤ sup Lπ V (t, x, i) +
π∈A1
j=1
h
i
0 ≤ sup V (t, (1 − κi πt )x, 0) − V (t, x, i) ,
π∈A2
d
h
i
X
0 = sup Lπ V (t, x, i) +
qi,j V (t, x, j)
π∈A1
j=1
h
i
· sup V (t, (1 − κi πt )x, 0) − V (t, x, i) .
π∈A2
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Verification for i > 0
Suppose both equations are equal to 0 at the same time. As before, we make
the ansatz
1
V (t, x, i) = x γ f i (t).
γ
With this, the equations reduce to
f i (t) = (1 − πti,∗ κi )γ f 0 (t),
∂ i
1
f (t) = −γf i (t) απti,∗ − (1 − γ)σ 2 (πti,∗ )2
∂t
2
Christoph Belak, Sören Christensen, Olaf Menkens
!
−
d
X
qi,j f j (t).
j=1
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Verification for i > 0
Suppose both equations are equal to 0 at the same time. As before, we make
the ansatz
1
V (t, x, i) = x γ f i (t).
γ
With this, the equations reduce to
f i (t) = (1 − πti,∗ κi )γ f 0 (t),
∂ i
1
f (t) = −γf i (t) απti,∗ − (1 − γ)σ 2 (πti,∗ )2
∂t
2
!
−
d
X
qi,j f j (t).
j=1
Taking the logarithm in the first equation and then the derivative with respect
to t, this yields
"
#
∂ i,∗
1
1 ∂ 0
1 ∂ i
i,∗ i
π =
(1 − πt κ ) 0
f (t) − i
f (t) .
∂t t
γκi
f (t) ∂t
f (t) ∂t
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Verification for i > 0
Plugging the ODEs for f 0 and f i into the last equation, we arrive at
"
2
∂ i,∗
1
1
i,∗ i
π = − i (1 − πt κ ) (1 − γ)σ 2 πti,∗ − πt0,∗
∂t t
κ
2
−
d
1X
q0,j (1 − π j,∗ κj )γ − 1
γ j=1
#
d
(1 − πtj,∗ κj )γ
1X
+
qi,j
,
γ j=1
(1 − πti,∗ κi )γ
with terminal condition πTi,∗ = 0.
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Overview
1
Introduction and Model Setup
2
The HJB System and Verification
3
Numerical Results
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Numerical Example
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Numerical Example
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Conclusions
We propose a regime switching model for an optimal investment problem in a
financial market with bubbles.
We derive a system of HJB equations which lead to a coupled system of
ordinary differential equations for the optimal strategies.
Depending on the choice of parameters, the optimal strategies may or may not
make the investor indifferent towards the impact of the crashes.
It is straightforward to extend the results to state-dependent market
coefficients.
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles
Introduction and Model Setup
The HJB System and Verification
Numerical Results
Thank you for your attention!!!
Christoph Belak, Sören Christensen, Olaf Menkens
Worst-Case Portfolio Optimization in a Market with Bubbles